cs.LGSep 28, 2026

ORPG: Reconciling Multiple Reward Objectives through Objective-wise Policy Gradients

Authors: Shicheng Fang, Yiwen Zhao, Wenbo Tian, Jiahao Lu, Yining Zheng, Yuxin Wang, Xipeng Qiu

Organizations: Fudan University · Shanghai Innovation Institute

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Abstract

Multi-reward policy optimization requires a joint update that reflects both the learning signals and the intended relationships among objectives. We introduce Objective-wise Reconciled Policy Gradient (ORPG), which constructs a separate clipped policy objective for each reward and reconciles the resulting gradients into one policy update. For compatible gradients, a cosine-dependent interpolation coordinates their contributions through a partially normalized reference while preserving the norm of their sum. We characterize this update as the unique solution of a spherical directional compromise. For conflicting gradients, projection follows the task's priorities. We evaluate the same compatible rule in helpfulness--safety alignment and correctness--cost optimization for mathematical reasoning. ORPG substantially improves average Useful and Harmless scores over the strongest external baseline on each axis. In mathematics, it achieves the highest average full-budget accuracy and three-budget hypervolume among the compared methods, with more accurate and shorter responses than the initial policy. Component comparisons and training dynamics show the larger contribution of compatible coordination and a complementary benefit from conflict handling. These results support gradient reconciliation for objectives with equal standing and for objectives with an explicit priority.

Figures & tables

Figure 1 : Overview of ORPG. Each reward retains its own clipped policy objective. Reconciliation uses the relationship between their full policy gradients to form a joint update.
MethodAlpacaHH-RLHFPKU-SafeRLHFAvg
UHUHUHUH
Base\text{2.536}_{\text{\tiny\pm0.025}}\text{2.941}_{\text{\tiny\pm0.028}}\text{2.855}_{\text{\tiny\pm0.002}}\text{3.977}_{\text{\tiny\pm0.002}}\text{4.644}_{\text{\tiny\pm0.002}}\text{6.404}_{\text{\tiny\pm0.004}}\text{3.345}_{\text{\tiny\pm0.009}}\text{4.441}_{\text{\tiny\pm0.010}}
GRPO\text{5.232}_{\text{\tiny\pm0.036}}\text{6.212}_{\text{\tiny\pm0.080}}\text{4.470}_{\text{\tiny\pm0.037}}\text{6.082}_{\text{\tiny\pm0.044}}\text{5.620}_{\text{\tiny\pm0.011}}\text{6.938}_{\text{\tiny\pm0.002}}\text{5.107}_{\text{\tiny\pm0.028}}\text{6.411}_{\text{\tiny\pm0.038}}
GDPO\text{5.335}_{\text{\tiny\pm0.010}}\text{6.250}_{\text{\tiny\pm0.121}}\text{4.546}_{\text{\tiny\pm0.004}}\text{6.116}_{\text{\tiny\pm0.019}}\text{5.640}_{\text{\tiny\pm0.008}}\text{6.939}_{\text{\tiny\pm0.005}}\text{5.174}_{\text{\tiny\pm0.005}}\text{6.435}_{\text{\tiny\pm0.047}}
GD 2 PO\text{5.332}_{\text{\tiny\pm0.010}}\text{6.297}_{\text{\tiny\pm0.033}}\text{4.541}_{\text{\tiny\pm0.007}}\text{6.138}_{\text{\tiny\pm0.010}}\text{5.631}_{\text{\tiny\pm0.001}}\text{6.939}_{\text{\tiny\pm0.002}}\text{5.168}_{\text{\tiny\pm0.005}}\text{6.458}_{\text{\tiny\pm0.013}}
ORPG\text{{5.943}}_{\text{\tiny\pm0.002}}\text{{7.061}}_{\text{\tiny\pm0.002}}\text{{5.044}}_{\text{\tiny\pm0.002}}\text{{6.679}}_{\text{\tiny\pm0.003}}\text{{5.781}}_{\text{\tiny\pm0.0001}}\text{{6.972}}_{\text{\tiny\pm0.0002}}\text{{5.589}}_{\text{\tiny\pm0.0004}}\text{{6.904}}_{\text{\tiny\pm0.001}}
Table 1 : Helpfulness–safety results. Each dataset reports Useful (U) and Harmless (H); Avg is the equal-weight average across datasets. Both scores are higher-is-better.
MethodA24AMCMATHMin.Oly.Avg
Acc (%)LenAcc (%)LenAcc (%)LenAcc (%)LenAcc (%)LenAcc (%)Len
Base\text{53.9}_{\text{\tiny\pm3.4}}\text{5698}_{\text{\tiny\pm60}}\text{{82.4}}_{\text{\tiny\pm0.7}}\text{3351}_{\text{\tiny\pm86}}\text{87.6}_{\text{\tiny\pm0.1}}\text{1462}_{\text{\tiny\pm6}}\text{37.3}_{\text{\tiny\pm0.3}}\text{1495}_{\text{\tiny\pm22}}\text{66.4}_{\text{\tiny\pm0.2}}\text{3821}_{\text{\tiny\pm9}}\text{65.5}_{\text{\tiny\pm0.7}}\text{3165}_{\text{\tiny\pm12}}
GRPO\text{42.2}_{\text{\tiny\pm1.0}}\text{2541}_{\text{\tiny\pm135}}\text{75.5}_{\text{\tiny\pm2.3}}\text{1708}_{\text{\tiny\pm58}}\text{86.0}_{\text{\tiny\pm0.2}}\text{939}_{\text{\tiny\pm40}}\text{37.9}_{\text{\tiny\pm0.6}}\text{964}_{\text{\tiny\pm91}}\text{61.8}_{\text{\tiny\pm0.3}}\text{1688}_{\text{\tiny\pm71}}\text{60.7}_{\text{\tiny\pm0.5}}\text{1568}_{\text{\tiny\pm66}}
GDPO\text{38.6}_{\text{\tiny\pm1.3}}\text{{2255}}_{\text{\tiny\pm105}}\text{74.7}_{\text{\tiny\pm0.6}}\text{{1540}}_{\text{\tiny\pm26}}\text{85.5}_{\text{\tiny\pm0.1}}\text{{833}}_{\text{\tiny\pm12}}\text{37.3}_{\text{\tiny\pm0.6}}\text{833}_{\text{\tiny\pm25}}\text{60.6}_{\text{\tiny\pm0.6}}\text{{1501}}_{\text{\tiny\pm46}}\text{59.3}_{\text{\tiny\pm0.1}}\text{{1392}}_{\text{\tiny\pm38}}
GD 2 PO\text{41.9}_{\text{\tiny\pm5.4}}\text{2470}_{\text{\tiny\pm90}}\text{75.1}_{\text{\tiny\pm1.5}}\text{1583}_{\text{\tiny\pm51}}\text{85.4}_{\text{\tiny\pm0.1}}\text{846}_{\text{\tiny\pm12}}\text{37.9}_{\text{\tiny\pm1.0}}\text{{826}}_{\text{\tiny\pm34}}\text{60.9}_{\text{\tiny\pm0.3}}\text{1553}_{\text{\tiny\pm29}}\text{60.3}_{\text{\tiny\pm0.8}}\text{1456}_{\text{\tiny\pm29}}
ORPG\text{{56.4}}_{\text{\tiny\pm1.9}}\text{4613}_{\text{\tiny\pm94}}\text{{82.4}}_{\text{\tiny\pm0.6}}\text{2748}_{\text{\tiny\pm39}}\text{{88.0}}_{\text{\tiny\pm0.1}}\text{1229}_{\text{\tiny\pm3}}\text{{39.3}}_{\text{\tiny\pm0.5}}\text{1287}_{\text{\tiny\pm27}}\text{{67.2}}_{\text{\tiny\pm0.6}}\text{2893}_{\text{\tiny\pm23}}\text{{66.7}}_{\text{\tiny\pm0.3}}\text{2554}_{\text{\tiny\pm22}}
Table 2 : Mathematical accuracy (Acc, %) and mean response length (Len, tokens) at the 8192-token budget. Each dataset is followed by the equal-weight Avg. A24: AIME-24; AMC: AMC-22-23; Min.: Minerva-Math; Oly.: OlympiadBench.
MethodA24AMCMATHMin.Oly.Avg
Base\text{0.280}_{\text{\tiny\pm0.015}}\text{0.608}_{\text{\tiny\pm0.006}}\text{0.768}_{\text{\tiny\pm0.001}}\text{0.325}_{\text{\tiny\pm0.002}}\text{0.485}_{\text{\tiny\pm0.0005}}\text{0.493}_{\text{\tiny\pm0.004}}
GRPO\text{0.311}_{\text{\tiny\pm0.011}}\text{0.620}_{\text{\tiny\pm0.014}}\text{0.769}_{\text{\tiny\pm0.003}}\text{0.336}_{\text{\tiny\pm0.003}}\text{0.507}_{\text{\tiny\pm0.007}}\text{0.509}_{\text{\tiny\pm0.003}}
GDPO\text{0.295}_{\text{\tiny\pm0.011}}\text{0.625}_{\text{\tiny\pm0.005}}\text{0.774}_{\text{\tiny\pm0.001}}\text{0.337}_{\text{\tiny\pm0.006}}\text{0.507}_{\text{\tiny\pm0.006}}\text{0.507}_{\text{\tiny\pm0.002}}
GD 2 PO\text{0.315}_{\text{\tiny\pm0.040}}\text{0.627}_{\text{\tiny\pm0.014}}\text{0.773}_{\text{\tiny\pm0.001}}\text{0.342}_{\text{\tiny\pm0.009}}\text{0.508}_{\text{\tiny\pm0.002}}\text{0.513}_{\text{\tiny\pm0.005}}
ORPG\text{{0.339}}_{\text{\tiny\pm0.010}}\text{{0.637}}_{\text{\tiny\pm0.004}}\text{{0.777}}_{\text{\tiny\pm0.0005}}\text{{0.344}}_{\text{\tiny\pm0.005}}\text{{0.516}}_{\text{\tiny\pm0.004}}\text{{0.523}}_{\text{\tiny\pm0.002}}
Table 3 : Mathematical hypervolume (HV, higher is better) over the 2048-, 4096-, and 8192-token budgets. Avg gives equal weight to each dataset.
UpdateUseful ↑\uparrowHarmless ↑\uparrow
ORPG\text{{5.589}}_{\text{\tiny\pm0.0004}}\text{{6.904}}_{\text{\tiny\pm0.001}}
Without compatible coordination\text{5.212}_{\text{\tiny\pm0.062}}\text{6.710}_{\text{\tiny\pm0.032}}
Without conflict resolution\text{5.563}_{\text{\tiny\pm0.0006}}\text{6.859}_{\text{\tiny\pm0.001}}
Without either component\text{5.193}_{\text{\tiny\pm0.073}}\text{6.695}_{\text{\tiny\pm0.085}}
CAGrad\text{5.183}_{\text{\tiny\pm0.002}}\text{6.626}_{\text{\tiny\pm0.002}}
Aligned-MTL\text{5.065}_{\text{\tiny\pm0.002}}\text{6.501}_{\text{\tiny\pm0.005}}
Table 4 : Reconciliation components and alternative gradient rules on helpfulness–safety. All rows retain separate policy objectives. Scores are averaged over the same three datasets as Table 1 .
Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

ParameterHelpfulness–safetyCorrectness–cost
Training steps100100
Batch / minibatch / rollout group512 / 128 / 4512 / 64 / 8
PPO epochs11
Learning rate2×10−62\times 10^{-6}10−610^{-6}
OptimizerAdamWAdamW
Schedule / warmupConstant / noneConstant / none
Appendix
Table 5 : Training settings. Token limits refer to training; evaluation budgets are specified separately.
Recorded operationSeconds per stepTotal hours
Total training step788.921.91
Rollout generation211.15.86
Actor update474.413.18
Old-policy log probabilities48.61.35
Reference-policy log probabilities46.41.29
Advantage computation2.40.07
Appendix
Table 6 : Recorded training-step costs for the default mathematical run on eight H200 GPUs. Component timers can overlap or nest.
DatasetUseCount
AlpacaTraining50,978
AlpacaReward calibration512
AlpacaEvaluation512
HH-RLHFEvaluation8,520
PKU-SafeRLHFEvaluation8,211
AIME-24Evaluation30
Appendix
Table 7 : Dataset sizes. Mathematical training uses DeepScaleR preview prompts.
Method204840968192
 Acc ↑\uparrowLen ↓\downarrowAcc ↑\uparrowLen ↓\downarrowAcc ↑\uparrowLen ↓\downarrow
Base\text{43.76}_{\text{\tiny\pm0.34}}\text{1434}_{\text{\tiny\pm2}}\text{53.66}_{\text{\tiny\pm0.48}}\text{2232}_{\text{\tiny\pm1}}\text{65.52}_{\text{\tiny\pm0.70}}\text{3165}_{\text{\tiny\pm12}}
GRPO\text{51.09}_{\text{\tiny\pm0.66}}\text{1299}_{\text{\tiny\pm54}}\text{{60.40}}_{\text{\tiny\pm0.48}}\text{1556}_{\text{\tiny\pm70}}\text{60.69}_{\text{\tiny\pm0.52}}\text{1568}_{\text{\tiny\pm66}}
GDPO\text{51.94}_{\text{\tiny\pm0.19}}\text{{1192}}_{\text{\tiny\pm19}}\text{59.00}_{\text{\tiny\pm0.41}}\text{{1385}}_{\text{\tiny\pm36}}\text{59.35}_{\text{\tiny\pm0.12}}\text{{1392}}_{\text{\tiny\pm38}}
GD 2 PO\text{{52.31}}_{\text{\tiny\pm1.27}}\text{1204}_{\text{\tiny\pm19}}\text{59.84}_{\text{\tiny\pm1.03}}\text{1439}_{\text{\tiny\pm33}}\text{60.26}_{\text{\tiny\pm0.85}}\text{1456}_{\text{\tiny\pm29}}
ORPG\text{47.04}_{\text{\tiny\pm0.32}}\text{1406}_{\text{\tiny\pm3}}\text{59.00}_{\text{\tiny\pm0.27}}\text{2076}_{\text{\tiny\pm8}}\text{{66.66}}_{\text{\tiny\pm0.26}}\text{2554}_{\text{\tiny\pm22}}
Appendix
Table 8 : Budget-specific measurements used to compute hypervolume. Accuracy (Acc) is in percent; length (Len) is in tokens.
UpdateAccuracy (%) ↑\uparrowLength ↓\downarrowHV ↑\uparrow
ORPG\text{{66.66}}_{\text{\tiny\pm0.26}}\text{2554.14}_{\text{\tiny\pm21.83}}\text{0.5226}_{\text{\tiny\pm0.0017}}
Without compatible coordination\text{66.15}_{\text{\tiny\pm0.19}}\text{2447.87}_{\text{\tiny\pm30.61}}\text{{0.5270}}_{\text{\tiny\pm0.0025}}
Without conflict resolution\text{65.46}_{\text{\tiny\pm0.55}}\text{2456.62}_{\text{\tiny\pm18.64}}\text{0.5203}_{\text{\tiny\pm0.0026}}
Without either component\text{66.04}_{\text{\tiny\pm0.14}}\text{{2399.56}}_{\text{\tiny\pm5.39}}\text{0.5236}_{\text{\tiny\pm0.0021}}
Appendix
Table 9 : Mathematical component comparisons. Accuracy and length use the 8192-token budget; HV aggregates the three budget points.
qqλ\lambdaUseful ↑\uparrowHarmless ↑\uparrow
0.250.25\text{5.5197}_{\text{\tiny\pm0.0005}}\text{6.8409}_{\text{\tiny\pm0.0012}}
0.50.25\text{{5.5891}}_{\text{\tiny\pm0.0004}}\text{{6.9038}}_{\text{\tiny\pm0.0014}}
0.750.25\text{5.5124}_{\text{\tiny\pm0.0004}}\text{6.8248}_{\text{\tiny\pm0.0011}}
0.50.125\text{5.4816}_{\text{\tiny\pm0.0017}}\text{6.8064}_{\text{\tiny\pm0.0015}}
0.50.5\text{5.5542}_{\text{\tiny\pm0.0012}}\text{6.8620}_{\text{\tiny\pm0.0009}}
Appendix
Table 10 : Sensitivity on helpfulness–safety. Scores are macro averages over the three evaluation sets.
qqλ\lambdaAccuracy (%) ↑\uparrowLength ↓\downarrowHV ↑\uparrow
0.250.25\text{65.58}_{\text{\tiny\pm0.33}}\text{2425.42}_{\text{\tiny\pm38.69}}\text{0.5227}_{\text{\tiny\pm0.0023}}
0.50.25\text{{66.66}}_{\text{\tiny\pm0.26}}\text{2554.14}_{\text{\tiny\pm21.83}}\text{0.5226}_{\text{\tiny\pm0.0017}}
0.750.25\text{66.60}_{\text{\tiny\pm0.51}}\text{2613.33}_{\text{\tiny\pm12.82}}\text{{0.5243}}_{\text{\tiny\pm0.0043}}
0.50.125\text{65.43}_{\text{\tiny\pm0.29}}\text{{2413.83}}_{\text{\tiny\pm14.08}}\text{0.5205}_{\text{\tiny\pm0.0028}}
0.50.5\text{65.52}_{\text{\tiny\pm0.49}}\text{2434.32}_{\text{\tiny\pm11.65}}\text{0.5218}_{\text{\tiny\pm0.0036}}
Appendix
Table 11 : Sensitivity on mathematics: full-budget accuracy and length, and three-budget hypervolume.
  204840968192
qqλ\lambdaAcc. ↑\uparrowLength ↓\downarrowAcc. ↑\uparrowLength ↓\downarrowAcc. ↑\uparrowLength ↓\downarrow
0.250.25\text{{47.77}}_{\text{\tiny\pm0.40}}\text{{1374.0}}_{\text{\tiny\pm1.7}}\text{59.31}_{\text{\tiny\pm0.44}}\text{{2015.2}}_{\text{\tiny\pm11.0}}\text{65.58}_{\text{\tiny\pm0.33}}\text{2425.4}_{\text{\tiny\pm38.7}}
0.50.25\text{47.04}_{\text{\tiny\pm0.32}}\text{1406.3}_{\text{\tiny\pm3.2}}\text{59.00}_{\text{\tiny\pm0.27}}\text{2075.9}_{\text{\tiny\pm7.9}}\text{{66.66}}_{\text{\tiny\pm0.26}}\text{2554.1}_{\text{\tiny\pm21.8}}
0.750.25\text{47.26}_{\text{\tiny\pm0.61}}\text{1390.6}_{\text{\tiny\pm3.2}}\text{59.33}_{\text{\tiny\pm0.36}}\text{2079.0}_{\text{\tiny\pm19.7}}\text{66.60}_{\text{\tiny\pm0.51}}\text{2613.3}_{\text{\tiny\pm12.8}}
0.50.125\text{47.34}_{\text{\tiny\pm0.49}}\text{1408.5}_{\text{\tiny\pm3.1}}\text{{60.55}}_{\text{\tiny\pm0.77}}\text{2035.7}_{\text{\tiny\pm14.5}}\text{65.43}_{\text{\tiny\pm0.29}}\text{{2413.8}}_{\text{\tiny\pm14.1}}
0.50.5\text{47.27}_{\text{\tiny\pm0.44}}\text{1380.7}_{\text{\tiny\pm2.0}}\text{59.44}_{\text{\tiny\pm0.21}}\text{2031.8}_{\text{\tiny\pm2.9}}\text{65.52}_{\text{\tiny\pm0.49}}\text{2434.3}_{\text{\tiny\pm11.6}}
Appendix
Table 12 : Mathematical sensitivity at each token budget. Accuracy is in percent and length in tokens.
UpdateStepsCosineConflictProjectionUseful RMS
ORPG1–330.6610.0000.0000.490
34–660.5840.0000.0000.500
67–1000.2650.1910.1910.857
Without compatible coordination1–330.6900.0000.0000.492
34–660.6160.0000.0000.488
67–1000.5220.0000.0000.507
Appendix
Table 13 : Training-stage gradient relationships and Useful advantage RMS. All versions use the same three step intervals.
MethodUsefulHarmlessUseful RMS
GDPO1.15681.4582–
GD 2 PO1.15771.4620–
Without either component1.15321.73680.5207
Without compatible coordination1.17381.76760.5278
Without conflict resolution1.49241.99900.8600
ORPG1.52172.00510.8868
Appendix
Table 14 : Late-training calibrated rewards and Useful advantage RMS (steps 81–100).

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